AD-A147 943 A LOCAL REFINEMENT FINITE ELEMENT METHOD FOR TIME i/iDEPENDENT PARTIAL DIFFE..(U) RENSSELAER POLYTECHNICINST TROY NY DEPT OF MATHEMATICAL SCIE.

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A]}OSR-TR. 8 4.-094 3SNAME Of PERFORMING ORGANIZATION OFFICE SYMBOL 7a. NAME OP MONITORING ORGANIZATION

Rensselaer Polytechnic firappicbirl Air Force Office of Scieni-ific Rasaaz-ch

6C. AOORESS #Cty State ad ZIP Codr, 7b. ADORESS iCet,. State and ZIP CodeDepartment of Computer Science Directorate of Mathematical and lnhformation

TroyNY 1180-590Sciences, Bolling AFB DC 20332do NtAME OF FUNOIkG/4PONSORING OFF SICE SYMBOL S. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER

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11 TiT%.F .Iatoe carsty Cljauutials,A LOCAL REFINEME~NT FINITE ELEMENT METHOD FOR TIME DEPENDENT PARTIAL D!EFFERENTIAL QUATI0NS

13, PERSONAL AUT"OR ISI

_Joseph E. Flaherty* and Peter K. Moore**13& TYPE Of REPORT j13b. TIME COVERED 1d VA T* OF REPORT (Yr. Me.. pa) 15. PAGE COUNT.Technical 1RO4M - TO ____AUG 84 11

IS. SUP PLEME1TARY' NOTATiON*and U.S. Army Armament, Munitions & Chemical Command Armament Research & Development Ctr,Large Calibre Weapon Systems Lab, Benet Weapons Lab, *atervliet NY 12189-5000 **ept ofMathematical Sciences, Rensselaer Polytechnic Inst, Troy NY 121.81.

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F TR- 84.9945 .

A LOCAL REFINEMENT FINITE ELEMENT METHOD FOR TIME DEPENDENTPARTIAL DIFFERENTIAL EQUATIONS 1

Joseph E. FlahertyDepartment of Computer ScienceRensselaer Polytechnic Institute

Troy, NY 12181 ijC Accession ForA4.1 NTIS GRA&I

and "" DTIC TAB 'Unannounced E -

U. S. Army Armament, Munitions, and Chemical Command Justir atio-.Armament Research and Development Center

Large Calibre Weapon Systems Laboratory ByBenet Weapons Laboratory Distribution/ ,Watervliet, NY 12189-5000

Availability Codes

-Try, NY 12181-----.

ABSTRACT. We discuss an adaptive local refinement finite element .~method for solving initial-boundary value problems for vector systems of

partial differential equations in one space dimension and time. The method ,'uses piecewise bilinear rectangular space-time finite elements. For each time "step, grids are automatically added to regions where the local discretization :

" : error is estimated as being larger than a prescribed tolerance. We discuss~~several aspects of our algorithm, including th tree structure that is used to .":i:" represent the finite element solution and grids, an error estimation technique, '

*i~ and initial and boundary conditions at coarse-fine mesh interfaces. We also i*.- present computational results for a simple linear hyperbolic problem, a :

i-" problem involving Burgers' equation, and a model combustion problem.

":.:1. INTRODUCTION. There is an ever increasing need to solve!... ."problems of greater complexity and a corresponding need for reliable and:;,,-: robust software tools to accurately and efficiently describe the phenomena. .".:: Adaptive techniques are good candidates for providing the computational

-;. methods and codes necessary to solve some of these difficult problems. Two.-.. x .. popular adaptive techniques are: (i) moving mesh methods, where a grid of a :, ~fixed number of finite difference cells or finite elements is moved so as to.. ". follow and resolve local nonuniformities in the solution, and (ii) local ";

,'- ': refinement methods, where uniform fine grids are added to coarser grids in* .. regions where the solution is not adequately resolved. A representative

~sample of both types of methods is contained in Babuska, Chandr,, and

.. ::' The authors were partially supported by the U. S. Air Force Office of N:- Scientific Research, Air Force Systems Command, USAF, under Grant Number

:' : AFOSR 30-0192 and the U. S. Army Research Office under Contract Number

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Flaherty (2]. Recently, Adjerid and Flaherty [1] deveaopea a relementmethod that combines mesh moving and refinement.

Herein, we discuss a local refinement finite element procedure forfinding numerical solutions of M-dimensional vector systems of partial Ndifferential equations having the form

Lu ut * f(x,t,u,ux ) - [D(x,tu)ux x 0,

a < x < b, t > 0, (1.1)

subject to the initial conditions

u(xO) = u4(x) , aS x S b , (1.2)

and appropriate boundary conditions so that the problem has a well posedsolution.

We discretize (1.1,2) for a time step using a finite element-Galerkinprocedure with piecewise bilinear approximations on a rectangular space-timenet. At the end of each time step we estimate the local discretization error,add finer subgrids of space-time elements in regions of high error, andrecursively solve the problem again in these regions. The process terminateswhen the error estimate on each grid is less than a prescribed tolerance. Theoriginal coarse space-time grid is then carried forward for the next time stepand the strategy is repeated. Our algorithm is discussed further in Flahertyand Moore [9] and some of this discussion is repeated in Section 2.

Berger (31 used a similar local refinement procedure to solve one-andtwo-dimensional hyperbolic systems. She used explicit finite differenceschemes to discretize the partial differential equations, while we use implicitfinite element techniques since we are primarily interested in parabolicproblems.

In addition to the discretization technique, the major numericalquestions that must be answered as part of the development of a localrefinement code are (i) the estimation of the discretization error and (ii) theappropriate initial and boundary conditions to apply at coarse-fin. meshinterfaces. Of course, Computer Science questions, such as which languageto use to describe and implement the various algorithms and what datastructures to use to represent and store the grids and solutions must also be I."answered. Our work in all of these areas is still far from complete and hereinwe only discuss our progress and thoughts on error estimation techniques,data structures, and interface conditions (cf. Section 2). In Section 3, wepresent the results of three examples that illustrate our method and thediscussion of Section 2, and in Section 4, we present some preliminaryconclusions and future plans.

2. FINITE ELEMENT ALGORITHM. We discretize equation (1.1) on astrip a < x < 0, p < t < q using a finite element-Galerkin method with auniform grid of N rectangular elements of size (0 - a)/N by (q - p). Werefer to this grid as R(3.5,p.q,N,f,s), where f and s are pointers to thefather and son grids discussed later. Each grid uses records to store theappropriate information. 9_

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We generate the discrete system on R(u,O,p,q,N,f,s) in the usualmanner; thus, we ap.proximate u by U(x,t) and select test functions V(x,t),where U and V are elements of a space of C' bilinear polynomials with respectto the grid R. We then take the inner product of equation (1.1) and V,replace u by U, and integrate any diffusive terms by parts to obtain

S[V T , vTf(xtUU VTD(xt,,U)U ]dxdt

R [V--

1vTD(x,t,U)U I dt 0 (2.2)P

Equation (2.2) must vanish for all bilinear functions V on the grid R. Theintegrals are approximated using a four point Gauss quadrature rule and theresulting nonlinear system is solved by Newton iteration (cf.,e.g., [7] foradditional details). Appropriate initial and boundary conditions for (2.2) arediscussed later in this section.

We describe our local refinement procedure for solving problem (1.1,2) "for one time step (tl,tl ) on a coarse grid with N' elements, i.e, onR(a,b,t',tI,NI,0,s) (where the pointer f = 0 signifies that this grid has nofather). To solve this problem we simply call the procedure "locref" with thearguments R(a,b,t',t ,N',O,s), tol, tsub for each coarse grid time interval.A pseudo-PASCAL description of the procedure "locref" is shown in Figure 1.

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procedure locref (R(a,O,p,q,N,f,s), tol, tsub)begin

Solve the finite element equations (2.2) on R(t,5,p,q,N,f,s);Estimate the error on R(iO,p,q,N,f,s);if error > tol then

begincalculate where error > tol and return the son grids;for j := 1 to tsub do

for i 1 to number of sons dobegin

p[j] p * (j-1)*(q-p)/tsub;q[j] p[j] * (q-p)/tsub;locref (R(a[i], B[i],p[j],q[j],N[i] ,

R(*,5,p,q,N,f,s) ,s[i],tol,tsub)end

endend;

Figure I. Algorithm for local refinement solution of (1.1,2) onR(a,5,p,q,N,f,s) with an error tolerance of tol and dividing the local timestep by tsub each time the error test is not satisfied.

The recu-sive algorithm locref sets up a tree structure of grids withR~a..t,: %%.,s) being te root noce and with the solution being

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generated by a preorder traversal of the tree at each local time step. Forexample, if the root grid is refined to give two subgrids and the time step ishalved, then the problem is solved on the first subgrid on its first time step,then on the second subgrid on the same time step, then this procedure isrepeated for the second time step. The error is estimated by Richardsonextrapolation, i.e., the space and time steps are halved and the problem issolved again on this new grid. The two solutions that are obtained at eachoriginal grid point are used to generate an error estimate. If this pointwiseestimate exceeds the tolerance "tol", finer grids are added as leaf nodes tothe tree. This procedure is similar to one used by Berger [3]; however,there are more economical error estimation strategies (cf., e.g., Bietermanand Babuska [5, 6]) which we are currently investigating.

In order to solve, the finite element system (2.2) we need to supplyinitial and boundary conditions. On any grid with p = 0, s = a, or S = bthese can be obtained from the initial condition (1.2) or prescribed boundaryconditions. However, artificial initial and boundary conditions must be created 0. %at all other coarse-fine mesh interfaces. This is a difficult and crucialproblem that is discussed for explicit finite difference methods by Berger[3, 4J; however, it is largely unanswered for finite element applications.Instabilities or incorrect solutions (cf. Example 1 of Section 3) can result ifinappropriate conditions are specified.

For initial conditions, two strategies immediately come to mind: (i)saving all fine grid data for propagation in time or (ii) interpolating the bestcoarse grid data to finer grids. We consider a blend of the two strategieswhich consists of saving the fine grid data down to a given level I in thetree and subsequently interpolating for finer grids. Each grid in the first 1 L%7.levels either has a linked list of the initial data directly associated with it oruses an initial data list of an ancestor grid. To find the value of the solutionat some new initial point, the coordinate of that point is sequentially comparedto values in the linked list until an interval containing the point is found sothat interpolation can be used. This is costly and we are investigating moreefficient procedures that use the natural ordering that already exists. Weused either piecewise linear interpolation or piecewise parabolic interpolationwith shape preserving splines developed by McLaughlin (10]. For each gridin the first I levels of the tree, a linked list is created to store the initialdata. We are studying several alternative ways of determining a proper valuefor X.

At the present time, we prescribe internal Dirichlet boundary conditionsby linearly interpolating from coarse to finer grids. A buffer zone of twoelements is added to each end of regions of high error that do not intersectthe boundaries x = a and b. If two buffer zones overlap or are separatedfrom one another by one element, the two grids are joined. Similarly, if thebuffer is only one element away from either a or b, that element is added tothe grid.

3. NUMERICAL EXAMPLES. An experimental code based on thealgorithms in Section 2 has been written in FORTRAN-77. We are testing iton several examples, some of these follow and others are presented in [9).All results were computed in double precision on an IBM 3081D computer.

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Example 1. In order to illustrate the importance of adequately resolvinginitial conditions at each time step we solve the linear hyperbolic initial valueproblem

t x 0:

((1/2)(cos(201i(x-O.45)) -1)

u(x,0) u(x) =0.35 < x < 0.7510 , otherwise

We solve this problem for one coarse time step of At 0.05, 10 elements on 0< x < 1, tol = 0.01. For small enough times the exact solution is u*(x-t). Ifinitial conditions are interpolated from the coarse to the fine grid, theoscillations are missed and an incorrect solution is computed, possibly withouta user realizing that there is anything wrong. However, saving Initial valuesfor the first 8 levels of the tree of grids calculates the correct solution to theprescribed accuracy. The incorrect -and correct solutions are shown att 0.05 in Figure 2.

Example 2. We solve the following problem for Burgers' equation:

ut 4 uu x =dux 0 0< x < 1 0 0< t < I -t --,x..

u(x,0) sinix , 0 < x < 1

u(0,t) u(1,t 0 , t > 0

We choose d = 0.00003, a coarse grid of 10 elements and At = 0.1, andpiecewise parabolic approximations for. the initial conditions with ) = 6. It iswell known, that .tbeso| tioa, of this-problem is a "pulse" that steepens as ittravels to the right until it forms a shock layer at x = 1. After a time ofO(1/d) the pulse dissipates and the. splution- decays to zero. We solve thisproblem for *ol = 0.01 and 0.001 and show the solutions at t = 0.4 in Figure3. The solution with the cruder tolerance is exhibiting some oscillations thatare within our bounds. Thesqe,.bownAw,-- m--ot-Vttble when the finer

. tolaraoce -it used to solve the problem.

Example 3. We solve the model combustion problem°, 4 .. 2,u-° =o < x< I, .0<t<, IUt xx2 , 0x 1,<~

u(x,0) = 0 , u(0,t)=0 , u (1,t) = 0

The exponential nonlinearity is typical in combustion problems havingArrhenwus -chemical kinetics. However, in this case the solution develops a"hot spot" at x = I and becomes infinite when t is approximately 0.85. Wechoose a coarse grid of 20 elements and At = 0.05. tol 2 0.001, and I a 6. InFigure 4 we show the computed solution U(x,t) as a function of x for t0.05, 0.6, and 0.8 and in Figure 5 we show the mesh that was used to solvethe problem. We see that the mesh is initially concentrated in the region nearx = 0 where the curvature of the solution is largest. As time progresses andthe curvature diminishes, excessive refinement is not necessary. Finally, asthe solution begins to "blow-up" our algorithm generates a fine mesh only inthe region near x 1.

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4. DISCUSSION AND CONCLUSIONS. We have briefly described anadaptive local refinement algorithm for solving time dependent partialdifferential equations. Even though this is very much a working algorithm,and not a production code, we are very encouraged by the preliminaryresults. We are investigating several possible ways of improving theefficiency and robustness of our algorithm. These include adding higherorder ,polynomial finite element approximations, adaptively changing the -..number of elements that are carried forward in the coarse grid at each coarsetime step, how to select the appropriate buffer length, adaptively determiningthe optimal number of levels of initital conditions to keep at coarse-fineinterfaces, and the best boundary conditions to apply at internal boundaries.We are encouraged by the performance of McLaughlin's [10] shape preservingparabolic splines; however, the entire area of interpolating from coarse tofine grids needs further study. We are also developing non-Dirichlet"natural" boundary conditions to use at coarse-fine mesh interfaces.

Finally, we are very interested in combining the moving mesh strategyof, e.g., [7, 8] with the present local refinement strategy and extending ourmethods to two and three dimensions.

References

1. ADJERID, S. AND FLAHERTY, J. E., A Moving Finite Element Method forTime Dependent Partial Differential Equations with Error Estimation andRefinement, in preparation, 1984.

2. BABUSKA, I., CHANDRA, J., AND FLAHERTY, J. E. (Eds.), AdaptiveComputational Methods for Partial Differential Equations, SIAM,Philadelphia, 1983.

3. BERGER, M. J., Adaptive mesh refinement for hyperbolic partialdifferential equations, Report No. STAN-CS-82-924, Department ofComputer Science, Stanford University, 1982.

4. BERGER, M. J., Stability of interfaces with mesh refinement, Report No.83-42, Institute for Computer Applications in Science and Engineering,NASA Langley Research Center, Hampton, 1983.

5. BIETERMAN, M. AND BABUSKA, I., The finite elememt method forparabolic equations, I. a posteriori error estimation, Numer. Math., Vol.40, pp 339-371, 1982.

6. BIETERMAN, M. AND BABUSKA, I., The finite elememt method forparabolic equations, II. a posteriori error estimation and adaptiveapproach, Numer. Math., Vol. 40, pp 373-406, 1982.

7. DAVIS, S. F. AND FLAHERTY, J. E., An adaptive finite element methodfor initial-boundary value problems for partial differential equations, SIAM ,.* .

J. Sci. Stat. Comput., Vol. 3, pp. 6-27, 1982

8. FLAHERTY, J. E., COYLE, J. M., LUDWIG, R., AND DAVIS, S. F.,Adaptive finite element methods for parabolic partial differential equations, VAdaptive Computational Wethods for Partial Differential Equations,Sabuska, I.. Chandra, J., and Flaherty, J. E. (Eds.), SIAM,

L .~ ,

Philadelphia, 1983.

9. FLAHERTY, J. E. AND MOORE, P. K., An adaptive local refinementfinite element method for parabolic partial differential equations, Proc. ofthe International Conference on Accuracy -- Estimates and AdaptiveRefinements in Finite Element Computations, Technical University of ..-

Lisbon, Lisbon, Vol. 2, pp. 139-152, 1984.

10. McLAUG4LIN, H. W., Shape preserving planar interpolation: analgorithm, IEEE Compurer Graphics and Applics., Vol. 3, pp 58-67, 1983.

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